3
$\begingroup$

Let k be an algebraically closed field of characteristic 0. How to prove that $M_2$(k), the k-algebra of 2 by  2 matrices over k, is not isomorphic to the group ring of any finite group G over k.

$\endgroup$
3
$\begingroup$

Any such group must have order 4 and any group of order 4 is abelian so its group ring must be commutative. However, $M_2(k)$ is not commutative.

$\endgroup$
3
$\begingroup$

The group ring of a finite group is simple iff the group is trivial.

Indeed, there is always a $k$-algebra morphism $kG\to k$, coming from the trivial representation which, if $kG$ is simple, must be injective: this is only possible if $\dim kG=1$, that is, if $G$ is trivial.

Now matrix algebras are simple, so you get what you want.

$\endgroup$
  • $\begingroup$ This shows that no group algebra is isomorphic to no matrix algebra of any size —except in the trivial case. $\endgroup$ – Mariano Suárez-Álvarez May 4 '13 at 18:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.